A3 - SURDS

A3: SURDS ● Four operations on surds, including rationalising the denominator ● Solving equations involving surds 1. ABCD is a trapezium where AB is parallel to DC. The area of the trapezium is . 12 + 11 3 𝑐 𝑚 2 cm and cm. 𝐴𝐵 = 8 − 48 𝐷𝐶 = 12 − 12 Find the perpendicular distance between and , leaving your answer 𝐴𝐵 𝐷𝐶 in the form , where and are integers. 𝑎 + 𝑏 3 𝑎 𝑏 [5] 2. A rectangular block has a square base of side cm and a height of h 10+ 2 ( ) cm. The volume of the rectangular block is . 52 + 28 5 ( ) 𝑐 𝑚 3 Without using a calculator , find the height of the rectangular block, in cm. Give your answer in the form cm where and are integers. 𝑎 + 𝑏 5 ( ) 𝑎 𝑏 [5] 3. It is given that , where and are integers. 𝑝 − 𝑞 5 = 12 3 + 5 𝑝 𝑞 Without using a calculator, find the value of and . 𝑝 𝑞 [6] 4. Simplify . 7 2− 3 8+ 32 [2] 5. A rectangle has an area of . Given that the length is cm, 11 − 7 𝑐 𝑚 2 3 + 7 find the breadth of the rectangle and express the breadth in the form , 𝑎 − 𝑏 7 where and are integers. 𝑎 𝑏 [4] 6. It is given that . Find in the form , 2 3( 𝑥 + 1 ) = 2 − 𝑥 𝑥 𝑎 3− 𝑏 where and are real numbers. 𝑎 𝑏 [4] 7. Show can be written in the form , where and 2 5− 3 ( ) 6 + 5 ( )3 + 5 𝑎 + 𝑏 5 𝑎 𝑏 are rational numbers. [4] 8. Solve the equation , where , giving your answer in its 𝑥 3 − 4 𝑥 = 3 𝑥 ≠ 0 simplest surd form. [4]

9. (a) Without using a calculator, express in the form , 2 3+ 2 ( ) 2 𝑎 + 𝑏 6 where and are integers. 𝑎 𝑏 [2] 9. (b) Hence, or otherwise, express in the form , 2 3+ 2 ( ) 2 3 + 6 𝑝 + 𝑞 6 where and are rational numbers. 𝑝 𝑞 [3] 10. It is given that . Find in the form , where 3( 𝑥 + 3 ) = 𝑥 + 17 𝑥 𝑎 + 𝑏 3 where and are integers. 𝑎 𝑏 [4] 11. Given that , find the values of the integers 𝑎 − 3 5 ( ) 2 + 5 ( )= 𝑏 − 2 5 and . 𝑎 𝑏 [3] 12. Solve , leaving your answer in the form , 𝑥 + 7− 3 2= 1 𝑚 + 𝑛 2 where and are integers. 𝑚 𝑛 [3] 13. Simplify , giving your answer in the form , 108 − 12 3 𝑘 3 where is an integer. 𝑘 [2] 14. Without using a calculator, solve the equation . 3 𝑥 − 2= 2 1 + 𝑥 2 ( ) [5]

A3: SURDS (MARKING SCHEME) 1. ABCD is a trapezium where AB is parallel to DC. The area of the trapezium is . 12 + 11 3 𝑐 𝑚 2 cm and cm. 𝐴𝐵 = 8 − 48 𝐷𝐶 = 12 − 12 Find the perpendicular distance between and , leaving your answer 𝐴𝐵 𝐷𝐶 in the form , where and are integers. 𝑎 + 𝑏 3 𝑎 𝑏 Area of trapezium = , where and are the length of parallel sides 1 2 ( 𝑎 + 𝑏 ) ℎ 𝑎 𝑏 12 + 11 3 = 1 2 8 − 48+